Given a chief factor H/K of a finite group G, we say that a subgroup A of G avoids H/K if H∩A=K∩A; if HA=KA, then we say that A covers H/K. If A either covers or avoids the chief factors of some given chief series of G, we say that A is a partial CAP-subgroup of G. Assume that G has a Sylow p-subgroup of order exceeding pk. If every subgroup of order pk, where k≥1, and every subgroup of order 4 (when pk=2 and the Sylow 2-subgroups are non-abelian) are partial CAP-subgroups of G, then G is p-soluble of p-length at most 1.

The main aim of this note is to give a criterion for a subgroup-closed formation to be solubly saturated, which we hope may provide a useful proving ground for outstanding questions about this family of formations.

In a previous paper, the authors have shown that Eilenberg’s variety theorem can be extended to more general structures, called formations. In this paper, we give a general method to describe the languages corresponding to saturated formations of groups, which are widely studied in group theory. We recover in this way a number of known results about the languages corresponding to the classes of nilpotent groups, soluble groups and supersoluble groups. Our method also applies to new examples, like the class of groups having a Sylow tower.

Let p be a prime. We say that class X of hyperfinite p-groups determines p-nilpotency locally if every finite group G with a Sylow p-subgroup P in X is p-nilpotent if and only if N_G(P) is p-nilpotent. The results of this paper improve a recent result of Kurdachenko and Otal and show that if a hyperfinite group G has a pronormal Sylow p-subgroup in X, then G is p-nilpotent if and only if N_G(P) is p-nilpotent provided that X is closed under taking subgroups and epimorphic images. If X is not closed under taking epimorphic images, we have to impose local p-solubility to G. In this case, the hypothesis of pronormality can be removed.

Lower subsets of an ordered semigroup form in a natural way an ordered semigroup. This lower set operator gives an analogue of the power operator already studied in semigroup theory. We present a complete description of the lower set operator applied to varieties of ordered semigroups. We also obtain large families of fixed points for this operator applied to pseudovarieties of ordered semigroups, including all examples found in the literature. This is achieved by constructing six types of inequalities that are preserved by the lower set operator. These types of inequalities are shown to be independent in a certain sense. Several applications are also presented, including the preservation of the period for a pseudovariety of ordered semigroups whose image under the lower set operator is proper.Read More: http://www.worldscientific.com/doi/10.1142/S021819671540010X

In this paper families of non-nilpotent subgroups covering the non-nilpotent part of a finite group are considered. An A_5-free group possessing one of these families is soluble, and soluble groups with this property have Fitting length at most three. A bound on the number of primes dividing the order of the group is also obtained.

We present an extension of Eilenberg’s variety theorem, a well-known result connecting algebra to formal languages. We prove that there is a bijective correspondence between formations of finite monoids and certain classes of languages, the formations of languages. Our result permits to treat classes of finite monoids which are not necessarily closed under taking submonoids, contrary to the original theory. We also prove a similar result for ordered monoids.